This interval therefore divides the octave, which is assumed to have the ratio 2:1, into 43 equal parts. Thus a meride represents one degree in 43-edo tuning.

One potential defect of using merides is that the familiar 12-edo semitone does not come out with an integer number of merides, since 43 is a prime number and therefore does not divide evenly by 12. Thus, the 12-EDO semitone is ~3.583333333, or exactly 3 & 7/12, merides.

Note that 7 * 43 = 301, so both of Saveur's units, the merides and heptamerides, did divide evenly.

The interval which functions as the 5th in 43-edo is nearly identical to the 1/5-comma meantone "5th":

1/5-comma meantone was first described by Abraham Verheyen in a letter to Simon Stevin, around 1600.

Below is a lattice which shows one pattern by which the plane of the [3 5] lattice is tiled by 43-edo:

Below is a table of intervals, including all 23 intervals which occur in a 12-tone subset of 1/5-comma meantone, and some of the 43-edo equivalents of those, as well as several others in JI and in various other meantones.

Below is a graphic showing a 7-limit "closest to 1/1" (by the taxicab metric) periodicity-block for 43-edo, in which three 5-limit planes are shown side-by-side; exponents of 7 are 0, 1, and -1 from left to right.

Below is a plot of the points in a 3-dimensional Monzo lattice of the same 7-limit "closest to 1/1" periodicity-block for 43-edo. In addition to the usual rectangular vertices, here i have also drawn lines connecting the doubled and trebled pitches (shown in darker grey in the "5-limit sheets" lattices above).

Below is a table and graphs which show the error of 43-edo in representing all ratios in the 11-odd-limit (i.e., those in Partch's tonality diamond), both the absolute error in cents, and the relative error as a percentage of one degree of 43-edo.

It can be seen that 43-edo's success at representing the 11-limit ratios can be categorized as follows:

good: 7/11 and 11/7

fair: 1/3, 3, 5, 1/5, 7/5, 5/7, 11/5, and 5/11

mediocre: 5/3, 3/5, 7, 1/7, 1/9, 9, 11, and 1/11

poor: 7/3, 3/7, 5/9, 9/5, 9/7, 7/9, 11/3, 3/11, 9/11, and 11/9

REFERENCES

Ellis, Alexander. 1885.
Appendix XX, in his translation of
Helmholtz, On the Sensations of Tone, p 437.
Dover reprint 1954.

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mérides calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = mérides

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